A marble is 12mm in diameter. What is the smallest height box that is 96mm square that can hold 200 marbles and still have the lid fit on the box without interference?
To determine the smallest height box that can accommodate the given number of marbles while fitting the lid without interference, we need to calculate the volume of the marbles and compare it to the volume of the box.
First, let's calculate the volume of a single marble. The volume of a sphere can be found using the formula:
V = (4/3) * π * r^3
where V is the volume and r is the radius of the sphere. Since the diameter of the marble is given (12mm), we can find the radius by dividing the diameter by 2:
r = 12mm / 2 = 6mm
Plugging the radius into the formula, we can calculate the volume of a single marble:
V_marble = (4/3) * π * (6mm)^3
Next, let's calculate the volume of 200 marbles:
V_200_marbles = 200 * V_marble
Now, let's calculate the volume of the box. The volume of a rectangular box can be found using the formula:
V_box = length * width * height
We are given that the box is a square with dimensions 96mm x 96mm. Therefore, the area of the square is:
A_square = length * width = 96mm * 96mm
Since the box is cubic (height, length, and width are equal), we can set the length and width to be equal to the square root of the area:
length = width = √(A_square) = √(96mm * 96mm)
Finally, we need to compare the volume of the marbles (200 marbles multiplied by the volume of a single marble) with the volume of the box:
If V_200_marbles < V_box, then the box can hold the marbles with the lid fitting without interference. If V_200_marbles > V_box, then we need a larger box.
By performing these calculations, we will be able to determine the smallest height of the box that can accommodate the given number of marbles.